I solved it myself. If the complementary graph is bipartite the original graph is basically $$${2}$$$ complete subgraphs with some edges between them. Let $$${N}$$$ be the number of nodes. If $$${N}$$$ is odd the answer is $$${{N}{-}{N}{mod}{2}}$$$. If the $$${2}$$$ subgraphs have any edges between them then the answer is $$${N}$$$. Else it depends whether or not both components have odd number of nodes, or both have even number. In the first case the answer is $$${{N}{-}{2}}$$$, in the latter it's $$${N}$$$. Thing is I made a mistake and the original problem doesn't actually boil down to this so this blog is pointless